TALLY: Tally Languages

TC0: Constant-Depth Threshold Circuits

The class of decision problems solvable by polynomial-size, constant-depth circuits with unbounded fanin, which can use AND, OR, and NOT gates (as in AC0) as well as threshold gates. A threshold gate returns 1 if at least half of its inputs are 1, and 0 otherwise.

TC0 circuits of depth 3 are strictly more powerful than TC0 circuits of depth 2 [HMP+93].

TC0 circuits of depth 3 and quasipolynomial size can simulate all of ACC0[Yao90].

There is a function in AC0 (explicitly given in [She08]), whose computation with TC0 circuits of depth 2 requires an exponential number of gates.

[NR97] give a candidate pseudorandom function family computable in TC0, that is secure assuming a subexponential lower bound on the hardness of factoring. (See also [NRR01] for an improvement of this construction, as well as [Kha93].)

One implication is that, assuming such a bound, there is no natural proof in the sense of [RR97] separating TC0 from P/poly. (It is important for this that a function family, and not just a candidate pseudorandom generator, is computable in TC0.) Another implication is that functions in TC0 are likely to be difficult to learn.

The permanent of a 0-1 matrix cannot be computed in uniform TC0[All99].

In a breakthrough result [Hes01] (building on [BCH86] and [CDL01]), integer division was shown to be in UD-uniform TC0. Indeed division is complete for this class under AC0 reductions.

The class of languages accepted by a BQP machine subject to the constraint that at every time step t, the machine's state is exponentially close to a tree state -- that is, a state expressible by a polynomial-size tree of additions and tensor products (together with complex constants and |0> and |1> leaf nodes).

More formally, a uniform classical polynomial-time algorithm generates a sequence of gates g(1), ... ,g(p(n)). Each g(t) can be either be selected from some finite universal basis of unitary gates (the choice turns out not to matter), or can be a 1-qubit measurement. When we perform a measurement, the state evolves to one of two possible pure states, with the usual probabilities, rather than to a mixed state. We require that the final gate g(p(n)) is a measurement of the first qubit. If at least one intermediate state was more than distance 2-Ω(n) away from the nearest state of tree size at most p(n), then the outcome of the final measurement is chosen adversarially; otherwise it is given by the usual Born probabilities. The measurement must return 1 with probability at least 2/3 if the input is in the language, and with probability at most 1/3 otherwise.

Defined in [Aar03b], where it was also shown that TreeBQP is
contained in the third level of PH, which might provide weak evidence that
TreeBQP does not equal BQP.

TREE-REGULAR: Regular Tree-Valued Languages

Same as REG, except that now the inputs are trees (say, binary trees) instead of strings. Each vertex is labeled with a symbol from a fixed alphabet. Evaluation begins at the leaves and proceeds to the root. The state of the finite automaton at each vertex v is a function of (1) the states at v's children (if any), and (2) the symbol at v. The tree is in the language if and only if the automaton is in an 'accept' state at the root.